Even in this enlightened age, the theory of distributions is a highly misunderstood and undervalued business. I guess that one can even say with impunity that it tends to turn (some) mathematical people away because of (1) its reputation for austerity, (2) its popularity with physicists and electrical engineers who then (ab)use it, or (3) a positive coefficient linear combination of these two. But there is no denying that, from analytic number theory to quantum mechanics and smack-dab in the middle of hard analysis, the theory of distributions is here to stay — with bells on.

All of this notwithstanding, learning it from the extant standard sources has always been a rather an austere affair. Indeed, Wikipedia’s article on the subject lists, e.g., Hörmander, Gel’fand-Shilov, Stein-Weiss, Rudin (Functional Analysis), Strichartz, and Trèves: a wonderful selection, but all pretty heavy going. Of course Rudin’s book is an established gem, but note that his aim is grander than merely to expound distribution theory as such. Hard analysts would probably eat up the books by Stein and Weiss, as well as Hörmander’s and Trèves’ opuses. But that still leaves out almost all (in a particularly perverse indiscreet (≠ non-discrete) measure) mathematicians, certainly including myself.

I have always wanted to look at Gel’fand-Shilov, but never have; and I know nothing about Strichartz book — etvoilá! C’est tout! (If Wikipedia doesn’t mention it, it surely doesn’t exist… Hah!). But it all matters not: I am ecstatic to have the book by Duistermaat and Kolk in my hands, as it is a gem.

Here is how it starts out:

In undergraduate physics a lecturer will be tempted to say on certain occasions: “Let δ(x) be a function on the line that equals 0 away from 0 and is infinite at 0 in such a way that the total integral is 1…” Such a function δ(x) is an object … one frequently would like to use, but of course there is no such function… All the same, it is important to realize what our lecturer is trying to accomplish: to describe an object in terms of the way it behaves when integrated against a function. It is for such purposes that the theory of distributions, or “generalized functions,” was created. … [I]t has revolutionized modern analysis.

By the way, it is fascinating to learn on p. xi of the book under review that the Fourier analytic rendering of (Dirac’s) d(x) goes back to none other than Euler: δ(x) = (1/2π) ∑ exp(inx), where n ∈ Z. And Duistermaat and Kolk add that Euler indeed found the sum to be 0 when x is away from 0.

They go on to note that their book

aims to be a thorough, yet concise and application-oriented, introduction to the theory of distributions that can be covered in one semester. These constraints forced us to make choices: we try to be rigorous but do not construct a complete theory that prepares the reader for all aspects and applications of distributions.

Fair enough. Indeed, it is precisely this pedagogical position that makes Distributions: Theory and Applications so attractive, and in many ways preferable to the more austere texts mentioned above, at least for a first, but very serious, introduction.

Thus, qua prerequisites, “[t]he amount of functional analysis that is needed in our treatment is reduced to a bare minimum: only … uniform boundedness is used, while the Hahn-Banach theorems are applied to give alternative proofs, with one exception, of results obtained by different methods.” Additionally, Duistermaat and Kolk opt for the Riemann integral instead of the Lebesgue integral. Again, fair enough (and quite reasonable, I think).

Here, very briefly, is a sketch of the book: the first 10 chapters are there to nail down foundational material, notation, &c. (the authors note that “[m]athematically sophisticated readers, having perused the first ten chapters, might prefer to proceed immediately to Chaps. 14 and 15”). To be sure, Ch’s 11–13 include coverage of e.g., convolution of distributions and (cool!) fractional integration and differentiation. Then, as hinted by the authors, we get to the meat: Fourier transforms, distribution kernels, Fourier series, Sobolev spaces, and so on.

Distributions: Theory and Applications is laden with examples and exercises and it is evident that Duistermaat and Kolk decided to pull out if not all, then most of the stops in the later sections. I guess this makes for a possible criticism of the book from a pedagogical perspective: it is my (possible mistaken) impression that the problem sets are supposed to proceed from easy to difficult as the problem-index increases — I base this on spot-checks, and quick ones at that, so: caveat — and note that the latter exercises are possibly too fruity for common consumption.

For instance, on p. 207 we find that exercise 14.42 has to do with nothing less than the metaplectic representation presented in the context of Lie algebras. This engenders dissonance vis à vis the authors’ claim, in the Preface, that “the reader is assumed to have merely a working knowledge of linear algebra and of multidimensional real analysis… while only a few of the problems also require some acquaintance with the residue calculus of complex analysis in one variable. In some cases, the notion of a group will be encountered mainly in the form of a (one-parameter) group of transformations acting on Rn.” It is true, strictly speaking, that problem 14.42 is focused on a one-parameter subgroup of GL(2), and, beyond this, Duistermaat and Kolk present lots of hints, an inter alia a mini-lecture on a snippet of Lie theory, references to Hermitian and skew-Hermitian operators’ appearances elsewhere in the exercise sets, etc. But it strikes me that a student, still scared of the rank-nullity theorem, would be entirely out of his depth here (and, more than likely, he’d already be scared away by the first appearance of the uniform boundedness principle): the prerequisites the authors have proposed (as mentioned above) are, well, far too optimistic, at least if the exercises are to be handled properly.

On the other hand, mathematical sophistication covers many anxieties: if a kid is truly interested and not easily cowed (these things seem to go hand in hand, no?) even something like no. 14.42 might have the beneficent effect of having him read on: on p. 208 (14.42 takes four pages), Duistermaat and Kolk zoom in on the theorem of David Shale and André Weil addressing the double (i.e. metaplectic) cover of the (Lie) group SL(2,R), and their discussion is utterly marvelous. The flavor is that of quantum physics, this being the context in which Shale first dealt with the projective representation (called the Weil representation by most number theorists, the oscillator representation by many physicists, and the Segal-Shale-Weil representation by ecumenically minded folks, e.g., Duistermaat and Kolk: see p. 209) whose 2-cocycle defines the aforementioned double cover. They talk about creation and annihilation operators, Rodrigues’ formula, and Hermite functions in connection with which they pose the reader the problem of showing the completeness of the set of such functions as part of their meeting the requirement for being a basis for L2(R). And here, again, the earlier dissonance appears: it looks to me like a lot more familiarity with (complex) analysis is required in order to do this problem justice.

Duistermaat and Kolk go on to show that Hermite functions are eigenfunctions of the Fourier transform and in their final summation note, among other things, that their preceding discussion “is essentially the ‘algebraic’ treatment of the harmonic oscillator in quantum mechanics.” It’s an excellent compact treatment of a major topic at the intersection of number theory and physics, for which reason the intrinsic worth of no. 14.42 and, by extension, of all of Distributions: Theory and Applications is greatly increased. But proper appreciation of such material, and doing justice to these sportier exercises, is really vouchsafed to more advanced and mature students than what the authors’ above remarks would suggest.

Along these lines of culturally highly evocative problems, here are some other (quasi-random) samples: no. 16.11 deals with the Féjer kernel, no. 16.23 is titled “Gamma distribution, Lipschitz formula, and Eisenstein series” (and, yes, these are the Eisenstein series from the theory of elliptic modular forms), no. 17.7 deals with the Schrödinger operator (i.e., the life’s blood of the Schrödinger wave equation), and no. 18.6 is titled “Distributions as boundary values of holomorphic functions.”

Distributions: Theory and Applications comes equipped with a long section containing solutions to selected exercises; this is the 21st chapter of the book, and, in light of my earlier criticism, this certainly mitigates the problem considerably. From another point of view — and one that I prefer — Distributions: Theory and Applications is much more than a textbook for a one-semester introduction to the indicated subject: the reader/student who hangs in, reads the text while filling the margins and a notebook or two (or three, or four) along the way, and wrestles with the exercises, will be more than ready for more advanced works in the area as well as the business of applying distributions in a number of nontrivial settings. Additionally he will have been exposed to a great deal of serious mathematics from, if you’ll pardon the pun, all over the spectrum.

Michael Berg is Professor of Mathematics at Loyola Marymount University in Los Angeles, CA.